Question

Determine whether f is even, odd, or neither. You may wish to use a graphing calculator or computer to check your answer visually.

84. \(f\left( x \right) = x\left| x \right|\)

Short answer

The function f is an odd function.

Step-by-step solution

Condition for even and odd function

The function \(f\) is said to be an even function if it satisfies the condition \(f\left( { - x} \right) = f\left( x \right)\). The function is an odd function if it satisfies the condition \(f\left( { - x} \right) = - f\left( x \right)\).

Check even or odd function

Replace \(x\) with \( - x\) in the function \(f\left( x \right) = x\left| x \right|\)to obtain\(f\left( { - x} \right)\)as shown below:

\(\begin{aligned}f\left( { - x} \right) &= x\left| x \right|\\f\left( { - x} \right) &= \left( { - x} \right)\left| {\left( { - x} \right)} \right|\\f\left( { - x} \right) &= - \left( {x\left| x \right|} \right)\\f\left( { - x} \right) &= - f\left( x \right)\end{aligned}\)

Since \(f\left( { - x} \right) = - f\left( x \right)\), so \(f\left( x \right)\) is an odd function.

Check the answer visually

The procedure to draw the graph of the above equation by using the graphing calculator is as follows:

To check the answer visually draw the graph of the function\(f\left( x \right) = x\left| x \right|\)by using the graphing calculator as shown below:

1. Open the graphing calculator. Select the “STAT PLOT” and enter the equation\(X|X|\)in the\({Y_1}\)tab.
2. Enter the “GRAPH” button in the graphing calculator.

Visualization of graph of the function \(f\left( x \right) = x\left| x \right|\) is shown below:

It is observed that graph of the function is symmetric about the origin, thus the function is an odd function.